We present a new general complex integral transform in this study. This complex transform properties are studied. The main problem is reduced to a simple algebraic equation using this complex integral transformation. By solving this algebraic equation and using the inverse of the general complex integral transform, the answer to this main problem can be found. Finally, the solution to linear higher order ordinary differential equation is found by using the new general complex integral transform. We also present and discuss a number of key real-life problems, such as pharmacokinetics and nuclear physics.

In this work, a new integral transform called the “Kuffi-Abbas-Jawad” KAJ integral transform has been proposed and applied to solve ordinary differential equations. The proposed KAJ integral transform is a modification of the well-known (Sadiq-Emad-Eman) SEE integral transform. It is a novel method that could be used to solve ordinary differential equations. The KAJ integral transform has been tested and proven for its excellent ability to solve homogeneous and non-homogeneous higher-order ordinary differential equations.

Many integral transformations have been proposed, tasted and proven to solve many applications in various scientific fields. The diversity of integral transformations comes from their unique ability to solve problems by transforming them from one domain where the solution is given by a complex mathematical procedure to another domain where simple algebraic methods can solve them. On this basis, many of these transformations and knowledge appeared on the field of real numbers and the field of complex numbers. The reason for the diversity of these transforms is the need for them in solving some mathematical equations and physical equations or that pertain to any branch of applied sciences, so they have proven their strength in solving differential and complementary equations and differential systems in addition to scientific applications. The Laplace Transform was one of the first transformations that contributed to the solution of mathematical equations, and then dozens of transformations appeared, which depended mainly on the Laplace Transform. This work demonstrates the temporal diversity of the integral transformations by presenting a group of integral transformations with their basic characteristics. We dealt with single and double integrals

Transformation and many other substitution methods have been used to solve non-linear differential fractional equations. In this present work, the homotopy perturbation method to solve the non-linear differential fractional equation with the help of He’s Polynomials is provided as the transformation plays an essential role in solving differential linear and non-linear equations. Here is the α-Sumudu technique to find the relevant results of the gas dynamics equation in fractional order. To calculate the non-linear fractional gas dynamical problem, a consumer method created on the new homotopy perturbation a-Sumudu transformation method (HP TM) is suggested. In the Caputo type, the derivative is evaluated. a-Sumudu homotopy perturbation technique and He’s polynomials are all incorporated in the HPSaTM. The availability of He’s polynomials could be used to conveniently manage the non-linearity. The suggested approach shows that the strategy is simple to implement and provides results that can be compared to the results gained from any other transformation technique.